↑

Properties

Label 126.7.c.a
Level $126$
Weight $7$
Character orbit 126.c
Analytic conductor $28.987$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [126,7,Mod(55,126)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(126, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.55");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 126.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(28.9868145361\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.211968.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 30x^{2} + 207 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + 32 q^{4} - 5 \beta_{2} q^{5} + ( - 21 \beta_{3} + 7 \beta_{2} + \cdots + 77) q^{7}+ \cdots + 32 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + 32 q^{4} - 5 \beta_{2} q^{5} + ( - 21 \beta_{3} + 7 \beta_{2} + \cdots + 77) q^{7}+ \cdots + ( - 77567 \beta_{3} - 1862 \beta_{2} + \cdots - 206976) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 128 q^{4} + 308 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 128 q^{4} + 308 q^{7} + 4440 q^{11} - 2688 q^{14} + 4096 q^{16} + 1536 q^{22} - 40584 q^{23} - 40700 q^{25} + 9856 q^{28} + 18264 q^{29} + 84000 q^{35} - 23192 q^{37} - 44696 q^{43} + 142080 q^{44} + 33792 q^{46} - 310268 q^{49} - 364800 q^{50} - 248616 q^{53} - 86016 q^{56} - 840192 q^{58} + 131072 q^{64} - 1293600 q^{65} - 434776 q^{67} + 1102080 q^{70} + 451608 q^{71} + 981504 q^{74} + 309624 q^{77} + 2092904 q^{79} - 2117760 q^{85} + 2334720 q^{86} + 49152 q^{88} - 1109472 q^{91} - 1298688 q^{92} + 190560 q^{95} - 827904 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 30x^{2} + 207 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} + 18\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{3} + 34\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{2} + 60 ) / 3 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{2} - 3\beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{3} - 60 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -9\beta_{2} + 51\beta_1 ) / 16 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\)
\(\chi(n)\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
− 4.38664i
4.38664i
3.27984i
− 3.27984i
−5.65685 0 32.0000 − 98.3767i 0 195.794 − 281.627i −181.019 0 556.502i
55.2 −5.65685 0 32.0000 98.3767i 0 195.794 + 281.627i −181.019 0 − 556.502i
55.3 5.65685 0 32.0000 − 204.749i 0 −41.7939 + 340.444i 181.019 0 − 1158.23i
55.4 5.65685 0 32.0000 204.749i 0 −41.7939 − 340.444i 181.019 0 1158.23i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.7.c.a 4
3.b odd 2 1 14.7.b.a ✓ 4
7.b odd 2 1 inner 126.7.c.a 4
12.b even 2 1 112.7.c.c 4
15.d odd 2 1 350.7.b.a 4
15.e even 4 2 350.7.d.a 8
21.c even 2 1 14.7.b.a ✓ 4
21.g even 6 2 98.7.d.b 8
21.h odd 6 2 98.7.d.b 8
24.f even 2 1 448.7.c.e 4
24.h odd 2 1 448.7.c.h 4
84.h odd 2 1 112.7.c.c 4
105.g even 2 1 350.7.b.a 4
105.k odd 4 2 350.7.d.a 8
168.e odd 2 1 448.7.c.e 4
168.i even 2 1 448.7.c.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.7.b.a ✓ 4 3.b odd 2 1
14.7.b.a ✓ 4 21.c even 2 1
98.7.d.b 8 21.g even 6 2
98.7.d.b 8 21.h odd 6 2
112.7.c.c 4 12.b even 2 1
112.7.c.c 4 84.h odd 2 1
126.7.c.a 4 1.a even 1 1 trivial
126.7.c.a 4 7.b odd 2 1 inner
350.7.b.a 4 15.d odd 2 1
350.7.b.a 4 105.g even 2 1
350.7.d.a 8 15.e even 4 2
350.7.d.a 8 105.k odd 4 2
448.7.c.e 4 24.f even 2 1
448.7.c.e 4 168.e odd 2 1
448.7.c.h 4 24.h odd 2 1
448.7.c.h 4 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 51600T_{5}^{2} + 405720000 \) acting on \(S_{7}^{\mathrm{new}}(126, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 51600 T^{2} + 405720000 \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( (T^{2} - 2220 T + 1227492)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 26266196601792 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 126735731638272 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 551809934313408 \) Copy content Toggle raw display
$23$ \( (T^{2} + 20292 T + 100711044)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 9132 T - 1357906716)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 86\!\cdots\!12 \) Copy content Toggle raw display
$37$ \( (T^{2} + 11596 T - 1847926364)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 28\!\cdots\!32 \) Copy content Toggle raw display
$43$ \( (T^{2} + 22348 T - 10521464924)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 12\!\cdots\!88 \) Copy content Toggle raw display
$53$ \( (T^{2} + 124308 T + 13614948)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 82\!\cdots\!48 \) Copy content Toggle raw display
$67$ \( (T^{2} + 217388 T + 11809058788)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 225804 T - 95443772508)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 26\!\cdots\!72 \) Copy content Toggle raw display
$79$ \( (T^{2} - 1046452 T + 268612291876)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 21\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 15\!\cdots\!52 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 39\!\cdots\!12 \) Copy content Toggle raw display
show more
show less